I've never played any game from this series and haven't been following it. It just happened that I was online when the stream started so I thought "what the heck-why not check it out?"
What I got from it is that you go running around in a group destroying monsters and stuff. Which is my idea of what MH is. Except the overworld in X is cooler. So what is the big difference? (Super n00b here).
EDIT:
I see somebody answered my question! =D
For a more detailed comparison:
Monster Hunter is all about knowing the monster's attack patterns down to how long do their animations last, (same with your own attacks, as you cannot cancel an attack once initiated) Frame animation is very important since you have to know when its the right moment to attack, evade, heal, etc. Monster Hunter's various attacks vary depending on the weapon and they are all performed real time, like an action game, each weapon has different combos and such.
Xenoblade (and by extension, Xenoblade X) uses a set of different "Arts" arts are the main attacks that each character has, with different effects and properties. For example Shulk has an art that does more damage if you position yourself behind and enemy and use it, while Reyn has an attack that does more damage if he uses it right after another one of his arts.
The characters in Xenoblade have automatic attacks, which are done automatically (duh) once you are close enough to the enemy, these do little damage, but have some uses like it helps fill up Shulk's Talent Gauge. Every character in Xenoblade had a different talent gauge, think of it as a special move, Riki could steal things, Melia restack her spells, Shulk got a different set of monado arts once activated, etc.
Every time an art is used there's a cooldown period where you have to wait to use it again, in X there's system that allows you to wait double the cooldown time to make your art more powerful when used again. Arts all have different properties, and some work best when combined with another. A common example is that one of Shulk's arts can inflict "break" status on an enemy, which allows them to be inflicted with the "Topple" status (makes them fall down) Shulk doesn't have any art that inflicts topple (unless he switches to monado arts with his talent gauge) but his most of his teamates do, so Shulk has to rely on his teamates and vice-versa to fully use the potential of everyone's arts.
So basically MH is an action game where knowing your own attacks and the monster's is the key (preparation with the right items is also very important) down to the animation of each attack, while Xenoblade's combat functions more like that of an RPG with some MMO elements. Both games emphasize teamwork (although the first xenoblade was singleplayer, so you had to rely on the AI)